Abstract Shear wave elasticity imaging with ultrasound is an exciting new diagnostic modality that holds great promise for noninvasively imaging the mechanical properties of soft tissue. Despite the signi?cant potential of ultrasound- based shear wave methods for imaging liver ?brosis and other important soft tissue pathologies, some signi?cant problems remain unsolved. In particular, present quantitative shear wave parameter estimation approaches neglect some fundamental physics, namely the combined effects of shear wave diffraction and power law attenua- tion. As a result, present shear wave elasticity imaging approaches yield different results with different ultrasound probes when applied to the same viscoelastic shear wave phantom. In addition, numerical validation of shear wave parameter estimation approaches is presently hindered by the available three-dimensional (3D) simulation models for shear waves in soft tissue, which all have some fundamental problems. For example, integer-order partial differential equations fail to accurately represent the attenuation of shear waves in soft tissue, which follows a power law. Although fractional calculus is ideal for modeling power laws, 3D numerical computations that numerically evaluate time-fractional derivatives are computationally prohibitive with ?nite difference or ?nite element methods, so alternative approaches are needed for 3D numerical simulations of shear waves in soft tissue. We propose to solve these problems through the completion of two speci?c aims. In Aim 1, we will validate new 3D fractional calculus simulation models that describe the effects of shear wave diffraction and power law attenuation in soft tissue, where the new models are amenable to 3D numerical computations on computers with a modest amount of memory, unlike existing fractional calculus models. The new fractional calculus models will also describe the power law attenuation for the entire range of parameters observed in soft tissue, in contrast to existing fractional calculus models, which all break down within the central range of observed values. In Aim 2, we will establish new fractional calculus models for shear wave parameter estimation which are based on an approximation to the 3D fractional calculus models that we are proposing for forward shear wave calculations. We propose to show, through evaluations performed with 3D simulated and measured shear wave data, that the proposed fractional calculus estimator yields much more accurate values for the shear elasticity than estimates obtained from present state-of-the-art time-of-?ight and k-space approaches. Thus, we propose to overcome several important de?ciencies of the present models for forward calculations and parameter estimation, where the proposed fractional calculus models are ultimately expected to achieve signi?cant improvements in the consistency of noninvasive shear wave parameter measurements that will provide translatable effects in practice for important clinical applications of shear wave elasticity imaging.